\(\int \log (\frac {-11+5 x}{5+76 x}) \, dx\) [241]

Optimal result

Integrand size = 14, antiderivative size = 35 \[ \int \log \left (\frac {-11+5 x}{5+76 x}\right ) \, dx=-\frac {1}{5} (11-5 x) \log \left (-\frac {11-5 x}{5+76 x}\right )-\frac {861}{380} \log (5+76 x) \]

[Out]

-1/5*(11-5*x)*ln((-11+5*x)/(5+76*x))-861/380*ln(5+76*x)

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2535, 31} \[ \int \log \left (\frac {-11+5 x}{5+76 x}\right ) \, dx=-\frac {1}{5} (11-5 x) \log \left (-\frac {11-5 x}{76 x+5}\right )-\frac {861}{380} \log (76 x+5) \]

[In]

Int[Log[(-11 + 5*x)/(5 + 76*x)],x]

[Out]

-1/5*((11 - 5*x)*Log[-((11 - 5*x)/(5 + 76*x))]) - (861*Log[5 + 76*x])/380

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 2535

Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*(B_.))^(p_.), x_Symbol] :> Simp[(a +
b*x)*((A + B*Log[e*((a + b*x)/(c + d*x))^n])^p/b), x] - Dist[B*n*p*((b*c - a*d)/b), Int[(A + B*Log[e*((a + b*
x)/(c + d*x))^n])^(p - 1)/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, A, B, n}, x] && NeQ[b*c - a*d, 0] && IGtQ
[p, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {1}{5} (11-5 x) \log \left (-\frac {11-5 x}{5+76 x}\right )-\frac {861}{5} \int \frac {1}{5+76 x} \, dx \\ & = -\frac {1}{5} (11-5 x) \log \left (-\frac {11-5 x}{5+76 x}\right )-\frac {861}{380} \log (5+76 x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.89 \[ \int \log \left (\frac {-11+5 x}{5+76 x}\right ) \, dx=\left (-\frac {11}{5}+x\right ) \log \left (\frac {-11+5 x}{5+76 x}\right )-\frac {861}{380} \log (5+76 x) \]

[In]

Integrate[Log[(-11 + 5*x)/(5 + 76*x)],x]

[Out]

(-11/5 + x)*Log[(-11 + 5*x)/(5 + 76*x)] - (861*Log[5 + 76*x])/380

Maple [A] (verified)

Time = 0.53 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.97

method	result	size
risch	\(x \ln \left (\frac {-11+5 x}{5+76 x}\right )-\frac {11 \ln \left (-11+5 x \right )}{5}-\frac {5 \ln \left (5+76 x \right )}{76}\)	\(34\)
parts	\(x \ln \left (\frac {-11+5 x}{5+76 x}\right )-\frac {11 \ln \left (-11+5 x \right )}{5}-\frac {5 \ln \left (5+76 x \right )}{76}\)	\(34\)
parallelrisch	\(x \ln \left (\frac {-11+5 x}{5+76 x}\right )-\frac {861 \ln \left (x -\frac {11}{5}\right )}{380}+\frac {5 \ln \left (\frac {-11+5 x}{5+76 x}\right )}{76}\)	\(40\)
derivativedivides	\(\frac {861 \ln \left (-\frac {861}{5+76 x}\right )}{380}+\frac {\ln \left (\frac {5}{76}-\frac {861}{76 \left (5+76 x \right )}\right ) \left (\frac {5}{76}-\frac {861}{76 \left (5+76 x \right )}\right ) \left (5+76 x \right )}{5}\)	\(44\)
default	\(\frac {861 \ln \left (-\frac {861}{5+76 x}\right )}{380}+\frac {\ln \left (\frac {5}{76}-\frac {861}{76 \left (5+76 x \right )}\right ) \left (\frac {5}{76}-\frac {861}{76 \left (5+76 x \right )}\right ) \left (5+76 x \right )}{5}\)	\(44\)

[In]

int(ln((-11+5*x)/(5+76*x)),x,method=_RETURNVERBOSE)

[Out]

x*ln((-11+5*x)/(5+76*x))-11/5*ln(-11+5*x)-5/76*ln(5+76*x)

Fricas [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.94 \[ \int \log \left (\frac {-11+5 x}{5+76 x}\right ) \, dx=x \log \left (\frac {5 \, x - 11}{76 \, x + 5}\right ) - \frac {5}{76} \, \log \left (76 \, x + 5\right ) - \frac {11}{5} \, \log \left (5 \, x - 11\right ) \]

[In]

integrate(log((-11+5*x)/(5+76*x)),x, algorithm="fricas")

[Out]

x*log((5*x - 11)/(76*x + 5)) - 5/76*log(76*x + 5) - 11/5*log(5*x - 11)

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.91 \[ \int \log \left (\frac {-11+5 x}{5+76 x}\right ) \, dx=x \log {\left (\frac {5 x - 11}{76 x + 5} \right )} - \frac {11 \log {\left (x - \frac {11}{5} \right )}}{5} - \frac {5 \log {\left (x + \frac {5}{76} \right )}}{76} \]

[In]

integrate(ln((-11+5*x)/(5+76*x)),x)

[Out]

x*log((5*x - 11)/(76*x + 5)) - 11*log(x - 11/5)/5 - 5*log(x + 5/76)/76

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.94 \[ \int \log \left (\frac {-11+5 x}{5+76 x}\right ) \, dx=x \log \left (\frac {5 \, x - 11}{76 \, x + 5}\right ) - \frac {5}{76} \, \log \left (76 \, x + 5\right ) - \frac {11}{5} \, \log \left (5 \, x - 11\right ) \]

[In]

integrate(log((-11+5*x)/(5+76*x)),x, algorithm="maxima")

[Out]

x*log((5*x - 11)/(76*x + 5)) - 5/76*log(76*x + 5) - 11/5*log(5*x - 11)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 139 vs. \(2 (30) = 60\).

Time = 0.32 (sec) , antiderivative size = 139, normalized size of antiderivative = 3.97 \[ \int \log \left (\frac {-11+5 x}{5+76 x}\right ) \, dx=-\frac {861 \, \log \left (\frac {\frac {5 \, {\left (\frac {5 \, {\left (5 \, x - 11\right )}}{76 \, x + 5} + 11\right )}}{\frac {76 \, {\left (5 \, x - 11\right )}}{76 \, x + 5} - 5} + 11}{\frac {76 \, {\left (\frac {5 \, {\left (5 \, x - 11\right )}}{76 \, x + 5} + 11\right )}}{\frac {76 \, {\left (5 \, x - 11\right )}}{76 \, x + 5} - 5} - 5}\right )}{76 \, {\left (\frac {76 \, {\left (5 \, x - 11\right )}}{76 \, x + 5} - 5\right )}} - \frac {861}{380} \, \log \left (\frac {{\left | 5 \, x - 11 \right |}}{{\left | 76 \, x + 5 \right |}}\right ) + \frac {861}{380} \, \log \left ({\left | \frac {76 \, {\left (5 \, x - 11\right )}}{76 \, x + 5} - 5 \right |}\right ) \]

[In]

integrate(log((-11+5*x)/(5+76*x)),x, algorithm="giac")

[Out]

-861/76*log((5*(5*(5*x - 11)/(76*x + 5) + 11)/(76*(5*x - 11)/(76*x + 5) - 5) + 11)/(76*(5*(5*x - 11)/(76*x + 5
) + 11)/(76*(5*x - 11)/(76*x + 5) - 5) - 5))/(76*(5*x - 11)/(76*x + 5) - 5) - 861/380*log(abs(5*x - 11)/abs(76
*x + 5)) + 861/380*log(abs(76*(5*x - 11)/(76*x + 5) - 5))

Mupad [B] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.83 \[ \int \log \left (\frac {-11+5 x}{5+76 x}\right ) \, dx=x\,\ln \left (\frac {5\,x-11}{76\,x+5}\right )-\frac {5\,\ln \left (x+\frac {5}{76}\right )}{76}-\frac {11\,\ln \left (x-\frac {11}{5}\right )}{5} \]

[In]

int(log((5*x - 11)/(76*x + 5)),x)

[Out]

x*log((5*x - 11)/(76*x + 5)) - (5*log(x + 5/76))/76 - (11*log(x - 11/5))/5